PX275 - G10 - plane waves

• the solution for the 3D wave equation:

where, ${\omega }^2=c^2|k{|}^2$

• considering two points on a plane perpendicular to $\overrightarrow{k}$, ${\overrightarrow{r}}_1$ and ${\overrightarrow{r}}_2$

• so, $\overrightarrow{k}\cdot ({\overrightarrow{r}}_2-{\overrightarrow{r}}_2)=0$

• the vector equation for a plane is: $$\vec k \cdot \vec r = d$$

• for a plane wave, $(\overrightarrow{k}\cdot +\omega t)$ is constant across a plane

• the planes represent a nominal phase of a wave

image: A-M Broomhall, lecture notes

• surfaces of constant phase are parallel planes

• 'right' moving waves have $\overrightarrow{k}$ in the positive direction, and 'left' moving in the negative

• therefore, a wavevector gives:

  • the direction of the plane wave
  • its wavelength: $\lambda =2\pi /|\overrightarrow{k}|$